Found problems: 85335
2020 Purple Comet Problems, 6
A given in finite geometric series with first term $a \ne 0$ and common ratio $2r$ sums to a value that is $6$ times the sum of an infi nite geometric series with first term $2a$ and common ratio $r$. Then $r = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2017 AMC 8, 21
Suppose $a$, $b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$?
$\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and }-1$
2011 Greece Team Selection Test, 4
Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.
2023 China Second Round, 7
We throw a dice three times and the numbers are $x,y,z$
find out the possibility of $\binom{7}{x}<\binom{7}{y}<\binom{7}{z}$
2020 Jozsef Wildt International Math Competition, W58
In all triangles $ABC$ does it hold that:
$$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
1984 USAMO, 5
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*}
P(0) = P(3) = \cdots &=& P(3n) = 2, \\
P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\
P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\
&& P(3n+1) = 730.\end{eqnarray*}
Determine $n$.
2023 Centroamerican and Caribbean Math Olympiad, 1
A [i]coloring[/i] of the set of integers greater than or equal to $1$, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible [i]colorings[/i] of the set of integers greater than or equal to $1$ that follow this rule.
2007 Hanoi Open Mathematics Competitions, 12
Calculate the sum
$\frac{5}{2.7}+\frac{5}{7.12}+...+\frac{5}{2002.2007}$
2014 IMS, 2
Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.
2009 Swedish Mathematical Competition, 1
Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.
2007 Germany Team Selection Test, 2
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]
2004 AIME Problems, 14
Consider a string of $n$ 7's, $7777\cdots77$, into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight 7's in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value 7000?
2011 Harvard-MIT Mathematics Tournament, 7
Let $A = \{1,2,\ldots,2011\}$. Find the number of functions $f$ from $A$ to $A$ that satisfy $f(n) \le n$ for all $n$ in $A$ and attain exactly $2010$ distinct values.
2025 CMIMC Geometry, 4
Let $ABCDEF$ be a regular hexagon with side length $1,$ and let $G$ be the midpoint of side $\overline{CD},$ and define $H$ to be the unique point on side $\overline{DE}$ such that $AGHF$ is a trapezoid. Find the length of the altitude dropped from point $H$ to $\overline{AG}.$
2014 Saudi Arabia Pre-TST, 4.4
Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.
2017 Regional Olympiad of Mexico Southeast, 3
Let $p$ of prime of the form $3k+2$ such that $a^2+ab+b^2$ is divisible by $p$ for some integers $a$ and $b$. Prove that both of $a$ and $b$ are divisible by $p$.
2015 Regional Competition For Advanced Students, 3
Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board.
Determine the least integer that can be reached at the end by an appropriate sequence of moves.
(Theresia Eisenkölbl)
2008 Canada National Olympiad, 2
Determine all functions $ f$ defined on the set of rational numbers that take rational values for which
\[ f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y,
\]
for each $ x$ and $ y$.
2011 NZMOC Camp Selection Problems, 1
Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$
.
2021 Science ON grade VI, 2
Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$?
[i] (Adapted from Archimedes 2011, Traian Preda)[/i]
2020 Princeton University Math Competition, B1
Runey is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Runese consist of anywhere between $1$ and $5$ letters, inclusive. As well, Runey can choose to add emphasis on any letter(s) that he chooses in a given word, hence making it a totally distinct word! What is the maximum number of possible words in Runese?
1957 Putnam, A2
Let $a>1.$ A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x$ for $x\in [0,a]$, and the line segment $y=e^a$ for $x\in [a-1,a].$ The wire is then suspended from the point $(a-1, e^a)$ and a horizontal force $F$ is applied to the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Show that the force $F$ is directed to the right.
1979 Putnam, A1
Find positive integers $n$ and $a_1, a_2, \dots, a_n$ such that $$a_1+a_2+\dots a_n=1979$$ and the product $a_1a_2\dots a_n$ as large as possible.
2007 Switzerland - Final Round, 1
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$
2024 Sharygin Geometry Olympiad, 8
Let $ABCD$ be a quadrilateral $\angle B = \angle D$ and $AD = CD$. The incircle of triangle $ABC$ touches the sides $BC$ and $AB$ at points $E$ and $F$ respectively. Prove that the midpoints of segments $AC, BD, AE,$ and $CF$ are concyclic.